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High Quality Content by WIKIPEDIA articles! In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gaussian curvature. In fact, one can find a metric with constant Gaussian curvature in any given conformal class. As a special case, every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to each other.